{"history":[],"article":{"id":"bc56eb02-7878-4e40-87ee-33f9d9413565","title":"Lorentz space","slug":"lorentz-space","content":":::summary\n:::\n\n# \nIn [mathematical analysis](https://bliptext.com/articles/mathematical-analysis), **Lorentz spaces**, introduced by [George G. Lorentz](https://bliptext.com/articles/george-g-lorentz) in the 1950s, are generalisations of the more familiar [L^{p} spaces](https://bliptext.com/articles/lp-space).\nThe Lorentz spaces are denoted by L^{p,q}.\nLike the L^{p} spaces, they are characterized by a [norm](https://bliptext.com/articles/norm-mathematics) (technically a quasi[norm](https://bliptext.com/articles/norm-mathematics)) that encodes information about the \"size\" of a function, just as the L^{p} [norm](https://bliptext.com/articles/norm-mathematics) does.\nThe two basic qualitative notions of \"size\" of a function are: how tall is the graph of the function, and how spread out is it.\nThe Lorentz norms provide tighter control over both qualities than the L^{p} norms, by exponentially rescaling the measure in both the range (p) and the domain (q).\nThe Lorentz norms, like the L^{p} norms, are invariant under arbitrary rearrangements of the values of a function.\n\n# Definition\nThe Lorentz space on a [measure space](https://bliptext.com/articles/measure-space) (X, \\mu) is the space of complex-valued [measurable functions](https://bliptext.com/articles/measurable-function) f on X such that the following [quasinorm](https://bliptext.com/articles/quasinorm) is finite\nwhere and.\nThus, when q < \\infty,\nand, when q = \\infty,\nIt is also conventional to set.\n\n# Decreasing rearrangements\nThe quasinorm is invariant under rearranging the values of the function f, essentially by definition.\nIn particular, given a complex-valued [measurable function](https://bliptext.com/articles/measurable-function) f defined on a measure space, (X, \\mu), its **decreasing rearrangement** function, can be defined as\nwhere d_{f} is the so-called **distribution function** of f, given by\nHere, for notational convenience, is defined to be \\infty.\nThe two functions |f| and f^{\\ast} are **equimeasurable**, meaning that\nwhere \\lambda is the [Lebesgue measure](https://bliptext.com/articles/lebesgue-measure) on the real line.\nThe related [symmetric decreasing rearrangement](https://bliptext.com/articles/symmetric-decreasing-rearrangement) function, which is also equimeasurable with f, would be defined on the real line by\nGiven these definitions, for and, the Lorentz quasinorms are given by\n\n# Lorentz sequence spaces\nWhen (the counting measure on \\mathbb{N}), the resulting Lorentz space is a [sequence space](https://bliptext.com/articles/sequence-space).\nHowever, in this case it is convenient to use different notation.\n\n## Definition.\nFor (or in the complex case), let denote the p-norm for and the ∞-norm.\nDenote by \\ell_p the Banach space of all sequences with finite p-norm.\nLet c_0 the Banach space of all sequences satisfying, endowed with the ∞-norm.\nDenote by c_{00} the normed space of all sequences with only finitely many nonzero entries.\nThese spaces all play a role in the definition of the Lorentz sequence spaces d(w,p) below.\nLet be a sequence of positive real numbers satisfying, and define the norm.\nThe Lorentz sequence space d(w,p) is defined as the Banach space of all sequences where this norm is finite.\nEquivalently, we can define d(w,p) as the completion of c_{00} under.\n\n# Properties\nThe Lorentz spaces are genuinely generalisations of the L^{p} spaces in the sense that, for any p,, which follows from [Cavalieri's principle](https://bliptext.com/articles/cavalieri-s-principle).\nFurther, coincides with [weak L^{p}](https://bliptext.com/articles/lp-space).\nThey are [quasi-Banach spaces](https://bliptext.com/articles/quasinorm) (that is, quasi-normed spaces which are also complete) and are normable for and.\nWhen p = 1, is equipped with a norm, but it is not possible to define a norm equivalent to the quasinorm of, the weak L^{1} space.\nAs a concrete example that the triangle inequality fails in, consider\nwhose quasi-norm equals one, whereas the quasi-norm of their sum f + g equals four.\nThe space L^{p,q} is contained in L^{p, r} whenever q < r. The Lorentz spaces are real [interpolation spaces](https://bliptext.com/articles/interpolation-space) between L^{1} and L^{\\infty}.\n\n## Hölder's inequality\nwhere, , , and.\n\n## Dual space\nIf (X,\\mu) is a nonatomic σ-finite measure space, then\n(i) for 0

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